options(encoding = 'UTF-8')
#Loading all the necessary packages
if (!require("xts")) install.packages("caret")
if (!require("CASdatasets")) install.packages("CASdatasets", repos = "http://cas.uqam.ca/pub/", type="source")
if (!require("caret")) install.packages("caret")
if (!require("ggplot2")) install.packages("ggplot2")
if (!require("mgcv")) install.packages("mgcv")
if (!require("dplyr")) install.packages("dplyr")
if (!require("gridExtra")) install.packages("gridExtra")
if (!require("visreg")) install.packages("visreg")
if (!require("MASS")) install.packages("MASS")
if (!require("plotrix")) install.packages("plotrix")
if (!require("rgeos")) install.packages("rgeos", type="source")
if (!require("rgdal")) install.packages("rgdal", type="source")
if (!require("xtable")) install.packages("xtable")
if (!require("maptools")) install.packages("maptools")
if (!require("scales")) install.packages("scales")
if (!require("data.table")) install.packages
if (!require("broom")) install.packages("broom")
if (!require("stringi")) install.packages("stringi")
require("CASdatasets")
require("ggplot2")
require("mgcv")
require("caret")
require("gridExtra")
require("dplyr")
require("visreg")
require("MASS")
require("plotrix")
require("rgdal")
require("rgeos")
require("xtable")
require("maptools")
require("scales")
require("data.table")
require("broom")
require("stringi")## Loading the dataset
require("CASdatasets")
data("freMTPLfreq")
# 'Keep it simple' Old style : freMTPLfreq = subset(freMTPLfreq,
# Exposure<=1 & Exposure >= 0 & CarAge<=25) With tidyverse (dplyr in fact)
# See cheatsheet
# https://github.com/rstudio/cheatsheets/blob/main/data-transformation.pdf
dataset = freMTPLfreq %>%
filter(Exposure <= 1 & Exposure >= 0 & CarAge <= 25)
saveRDS(dataset, file = "../dataset.Rds")A good idea is to check whether the dataset has been loaded correctly. To do this, the following tools can be used:
head(dataset)## PolicyID ClaimNb Exposure Power CarAge DriverAge
## 1 1 0 0.09 g 0 46
## 2 2 0 0.84 g 0 46
## 3 3 0 0.52 f 2 38
## 4 4 0 0.45 f 2 38
## 5 5 0 0.15 g 0 41
## 6 6 0 0.75 g 0 41
## Brand Gas Region Density
## 1 Japanese (except Nissan) or Korean Diesel Aquitaine 76
## 2 Japanese (except Nissan) or Korean Diesel Aquitaine 76
## 3 Japanese (except Nissan) or Korean Regular Nord-Pas-de-Calais 3003
## 4 Japanese (except Nissan) or Korean Regular Nord-Pas-de-Calais 3003
## 5 Japanese (except Nissan) or Korean Diesel Pays-de-la-Loire 60
## 6 Japanese (except Nissan) or Korean Diesel Pays-de-la-Loire 60
str(dataset)## 'data.frame': 410864 obs. of 10 variables:
## $ PolicyID : Factor w/ 413169 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
## $ ClaimNb : int 0 0 0 0 0 0 0 0 0 0 ...
## $ Exposure : num 0.09 0.84 0.52 0.45 0.15 0.75 0.81 0.05 0.76 0.34 ...
## $ Power : Factor w/ 12 levels "d","e","f","g",..: 4 4 3 3 4 4 1 1 1 6 ...
## $ CarAge : int 0 0 2 2 0 0 1 0 9 0 ...
## $ DriverAge: int 46 46 38 38 41 41 27 27 23 44 ...
## $ Brand : Factor w/ 7 levels "Fiat","Japanese (except Nissan) or Korean",..: 2 2 2 2 2 2 2 2 1 2 ...
## $ Gas : Factor w/ 2 levels "Diesel","Regular": 1 1 2 2 1 1 2 2 2 2 ...
## $ Region : Factor w/ 10 levels "Aquitaine","Basse-Normandie",..: 1 1 8 8 9 9 1 1 8 6 ...
## $ Density : int 76 76 3003 3003 60 60 695 695 7887 27000 ...
summary(dataset)## PolicyID ClaimNb Exposure Power
## 1 : 1 Min. :0.00000 Min. :0.002732 f :95432
## 2 : 1 1st Qu.:0.00000 1st Qu.:0.200000 g :90663
## 3 : 1 Median :0.00000 Median :0.530000 e :76784
## 4 : 1 Mean :0.03925 Mean :0.559997 d :67660
## 5 : 1 3rd Qu.:0.00000 3rd Qu.:1.000000 h :26558
## 6 : 1 Max. :4.00000 Max. :1.000000 j :17978
## (Other):410858 (Other):35789
## CarAge DriverAge
## Min. : 0.000 Min. :18.0
## 1st Qu.: 3.000 1st Qu.:34.0
## Median : 7.000 Median :44.0
## Mean : 7.413 Mean :45.3
## 3rd Qu.:12.000 3rd Qu.:54.0
## Max. :25.000 Max. :99.0
##
## Brand Gas
## Fiat : 16653 Diesel :205299
## Japanese (except Nissan) or Korean: 79031 Regular:205565
## Mercedes, Chrysler or BMW : 19087
## Opel, General Motors or Ford : 37287
## other : 9738
## Renault, Nissan or Citroen :216684
## Volkswagen, Audi, Skoda or Seat : 32384
## Region Density
## Centre :159426 Min. : 2
## Ile-de-France : 69576 1st Qu.: 67
## Bretagne : 41986 Median : 288
## Pays-de-la-Loire : 38541 Mean : 1987
## Aquitaine : 31211 3rd Qu.: 1414
## Nord-Pas-de-Calais: 27111 Max. :27000
## (Other) : 43013
If one needs some help on a function, typing a question mark and the name of the function in the console opens the help file of the function. For instance,
?headWe will now have a descriptive analysis of the portfolio. The different variables available are PolicyID, ClaimNb, Exposure, Power, CarAge, DriverAge, Brand, Gas, Region, Density.
The variable PolicyID related to a unique identifier of the policy. We can check that every policy appears only once in the dataset
length(unique(dataset$PolicyID)) == nrow(dataset)## [1] TRUE
Another possibility is to check the frequency of each PolicyID using the function table. The result is a table that shows for each PolicyID how many lines are in the dataset. We can then use a second time the function table in this result to show the frequency. We expect to have only ones (with possibily zeros), meaning each PolicyID has a unique line.
table(table(dataset$PolicyID))##
## 0 1
## 2305 410864
To what corresponds the 0 ? It appears that in this dataset the variable PolicyID is a factor. A factor variable has different levels. It appears that some PolicyID may be missing here (removed from the dataset ?). It is as if we had a 3-level categorical variable, for instance, color of a car, which takes three possible values: red, blue, gray, but in our dataset, we would only have red and blue cars. Gray would still be a level, but with no observation (i.e. no row) corresponding to a gray car. To remove unused levels, we can rely on the function droplevels.
The Exposure reveals the fraction of the year during which the policyholder is in the portfolio. We can compute the total exposure by summing the policyholders’ exposures. Here we find 230 082.6 years.
We can show the number of months of exposure on a table. The function cut allows to categorize (bin) a numerical variable. We can specify where to ‘break’ and give a name to each level using the labels argument. The output is a factor variable.
table(cut(dataset$Exposure, breaks = seq(from = 0, to = 1, by = 1/12), labels = 1:12))##
## 1 2 3 4 5 6 7 8 9 10
## 62633 29216 33452 24213 19463 29565 18835 14438 21518 13653
## 11 12
## 12422 131456
Using the function prop.table, it is possible to represent this information in relative terms show the number of months of exposure on a table.
Exposures_prop = prop.table(table(cut(dataset$Exposure, breaks = seq(from = 0,
to = 1, by = 1/12), labels = 1:12)))
round(Exposures_prop, 4)##
## 1 2 3 4 5 6 7 8 9 10
## 0.1524 0.0711 0.0814 0.0589 0.0474 0.0720 0.0458 0.0351 0.0524 0.0332
## 11 12
## 0.0302 0.3200
Alternatively, we can use a barplot !
ggplot(dataset) + geom_bar(aes(x = cut(Exposure, breaks = seq(from = 0, to = 1,
by = 1/12), labels = 1:12))) + scale_x_discrete(name = "Number of months") +
scale_y_continuous(name = "Number of Policies", label = label_number()) +
ggtitle("Exposure in months")
What if we also want to show the percentage on the bars ?
ggplot(dataset, aes(x = cut(Exposure, breaks = seq(from = 0, to = 1, by = 1/12),
labels = 1:12), label = scales::percent(prop.table(stat(count)), accuracy = 0.1))) +
geom_bar() + geom_text(stat = "count", vjust = -0.5, size = 3) + scale_x_discrete(name = "Number of months") +
scale_y_continuous(name = "Number of Policies", label = label_number()) +
ggtitle("Exposure in months")Note that a barplot is used to plot factor variables (categorical variables). In our case, we categorized the variable Exposure using the function cut. If we do not want to categorize this variable, we should use a histogram. We can specify the number of bins (= 12) or the binwidth (= 1/12).
ggplot(dataset, aes(x=Exposure))+geom_histogram(binwidth =1/12, fill='gray', color='white') +
scale_x_continuous(name = "Exposure in fraction of years", breaks=seq(0,1,1/12), labels = round(seq(0,1,1/12), 3))+
scale_y_continuous(name = 'Number of Polices', labels = label_number()) +
ggtitle("Exposure in fraction of years")If you are not familiar with ggplot, I could recommend this cheat-sheet: https://github.com/rstudio/cheatsheets/blob/main/data-visualization-2.1.pdf
ggplot(dataset, aes(x = ClaimNb)) + geom_bar() + geom_label(stat = "count",
aes(label = percent(prop.table(after_stat(count)), accuracy = 0.01)), vjust = 0.5) +
scale_x_continuous(name = "Number of Claims") + scale_y_continuous(name = "Number of Polices",
labels = label_number()) + ggtitle("Proportion of policies by number of claims")We can compute the average claim frequency in this portfolio, taking into account the different exposures.
scales::percent(sum(dataset$ClaimNb)/sum(dataset$Exposure), accuracy = 0.01)Here, we obtain 7.01%.
Let us now look at the other variables.
The variable Power is a categorized variable, related to the power of the car. The levels of the variable are ordered categorically. We can see the different levels of a factor by using the function level in R:
levels(dataset$Power)## [1] "d" "e" "f" "g" "h" "i" "j" "k" "l" "m" "n" "o"
We can see the number of observations in each level of the variable, by using the function table.
table(dataset$Power)##
## d e f g h i j k l m n o
## 67660 76784 95432 90663 26558 17398 17978 9270 4593 1758 1276 1494
Remember however, that in insurance, exposures may differ from one policyholder to another. Hence, the table above, does NOT measure the exposure in each level of the variable Power. We can use the functions group_by and summarise from package dplyr to give us the exposure in each level of the variable. Check out the cheatsheet https://github.com/rstudio/cheatsheets/blob/main/data-transformation.pdf
Power.summary = dataset %>%
group_by(Power) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure))We can show this on a plot as well:
ggplot(Power.summary, aes(x = Power, y = totalExposure, fill = Power, color = Power,
label = scales::number(totalExposure))) + geom_bar(stat = "identity") +
geom_text(stat = "identity", vjust = -0.5) + scale_y_continuous(name = "Exposure in years",
labels = scales::number) + theme(legend.position = "none")Let us now look at the observed claim frequency in each level
Power.summary = dataset %>%
group_by(Power) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Frequency = sum(ClaimNb)/sum(Exposure))
Power.summaryWe can compute the ratio to the portfolio claim frequency.
portfolio.cf = sum(dataset$ClaimNb)/sum(dataset$Exposure)
# Can also be written as
portfolio.cf = with(dataset, sum(ClaimNb)/sum(Exposure))
ggplot(Power.summary, aes(x = Power, y = Obs.Claim.Frequency, color = Power,
fill = Power, label = percent(Obs.Claim.Frequency, accuracy = 0.01))) +
geom_bar(stat = "identity") + geom_hline(aes(yintercept = portfolio.cf),
color = "black", size = 2, linetype = "dashed", alpha = 0.33) + geom_label(vjust = -0.21,
fill = "white", alpha = 0.25) + annotate(geom = "text", x = "m", y = portfolio.cf,
vjust = -0.5, label = paste("Average claim freq. of portfolio: ", percent(portfolio.cf,
accuracy = 0.01)), color = "black") + scale_y_continuous(name = "Observed Claim Frequency",
labels = percent_format(accuracy = 0.01)) + theme(legend.position = "none")The vehicle age, in years. This is the first continuous variable that we encounter (although it only takes discrete values).
ggplot(dataset, aes(x = CarAge)) + geom_bar() + scale_x_continuous(name = "Age of the Car",
breaks = seq(0, 100, 5)) + scale_y_continuous(name = "Number of Polices",
labels = label_number())
Alternatively, we can use a histogram, with a binwidth of 1.
ggplot(dataset, aes(x = CarAge)) + geom_histogram(binwidth = 1, color = "black",
fill = "white") + scale_x_continuous(name = "Age of the Car", breaks = seq(0,
100, 5)) + scale_y_continuous(name = "Number of Polices", labels = label_number())Again, here, the exposures are not considered on the barplot/histogram. We can use ddply to correct this.
CarAge.summary = dataset %>%
group_by(CarAge) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure))
CarAge.summaryThen, we can plot the data onto a barplot, as before.
ggplot(CarAge.summary, aes(x = CarAge, y = totalExposure, fill = factor(CarAge),
color = factor(CarAge), label = label_number(accuracy = 1)(totalExposure))) +
geom_bar(stat = "identity") + geom_text(stat = "identity", color = "black",
hjust = 0.25, vjust = 0.5, angle = 45, check_overlap = TRUE) + scale_x_continuous(breaks = seq(0,
100, 5)) + scale_y_continuous(name = "Exposure in years", labels = label_number()) +
theme(legend.position = "none")We can see a large difference, specially for new cars, which makes sense ! Indeed, let us look at the Exposure for recent vehicles, using a boxplot for instance.
ggplot(dataset[dataset$CarAge < 5, ], aes(x = CarAge, y = Exposure, group = CarAge)) +
geom_boxplot() + ggtitle("Exposure of recent cars")Let us now also compute the claim frequency by age of car,
CarAge.summary = dataset %>%
group_by(CarAge) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
CarAge.summaryand plot it!
portfolio.cf = with(dataset, sum(ClaimNb)/sum(Exposure))
ggplot(CarAge.summary, aes(x = CarAge, y = Obs.Claim.Freq, label = percent(Obs.Claim.Freq,
accuracy = 0.01))) + geom_bar(stat = "identity") + geom_hline(yintercept = portfolio.cf,
color = "black", size = 2, linetype = "dashed", alpha = 0.33) + annotate(geom = "text",
x = 20, y = portfolio.cf, vjust = -0.5, label = paste("Average claim freq. of portfolio: ",
percent(portfolio.cf, accuracy = 0.01)), color = "black") + scale_x_continuous(name = "Age of the Car",
breaks = seq(0, 100, 5)) + scale_y_continuous(name = "Observed Claim Frequency",
labels = percent_format(accuracy = 0.01)) + theme(legend.position = "none")Similarly to the Age of the Car, we can visualize the Age of the Drivers.
DriverAge.summary = dataset %>%
group_by(DriverAge) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
head(DriverAge.summary, 9)We can show the Exposures by Age of the Driver
ggplot(DriverAge.summary, aes(x = DriverAge, y = totalExposure)) + geom_bar(stat = "identity",
width = 0.8) + scale_y_continuous(name = "Exposure in years", labels = label_number()) +
scale_x_continuous(name = "Age of the Driver", breaks = seq(10, 150, 10))and the observed claim frequency by Age.
ggplot(DriverAge.summary, aes(x = DriverAge, y = Obs.Claim.Freq)) + geom_line() +
geom_point() + scale_y_continuous(name = "Observed Claim Frequency", labels = percent,
breaks = seq(0, 0.5, 0.05)) + scale_x_continuous(name = "Age of the Driver",
breaks = seq(10, 150, 10))The variable Brand is a categorized variable, related to the brand of the car. We can see the different levels of a factor by using the function level in R:
levels(dataset$Brand)## [1] "Fiat"
## [2] "Japanese (except Nissan) or Korean"
## [3] "Mercedes, Chrysler or BMW"
## [4] "Opel, General Motors or Ford"
## [5] "other"
## [6] "Renault, Nissan or Citroen"
## [7] "Volkswagen, Audi, Skoda or Seat"
Brand.summary = dataset %>%
group_by(Brand) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Brand.summary| Brand | totalExposure | Number.Observations | Number.Claims | Obs.Claim.Freq |
|---|---|---|---|---|
| Fiat | 9464.32 | 16653 | 714 | 0.07544 |
| Japanese (except Nissan) or Korean | 31228.97 | 79031 | 2078 | 0.06654 |
| Mercedes, Chrysler or BMW | 10392.17 | 19087 | 828 | 0.07968 |
| Opel, General Motors or Ford | 21733.56 | 37287 | 1731 | 0.07965 |
| other | 5676.08 | 9738 | 412 | 0.07259 |
| Renault, Nissan or Citroen | 133460.24 | 216684 | 8905 | 0.06672 |
| Volkswagen, Audi, Skoda or Seat | 18127.23 | 32384 | 1459 | 0.08049 |
ggplot(Brand.summary, aes(x = reorder(Brand, totalExposure), y = totalExposure,
fill = Brand, label = label_number()(totalExposure))) + geom_bar(stat = "identity") +
coord_flip() + guides(fill = "none") + scale_x_discrete(name = "") + scale_y_continuous("Exposure in years",
labels = label_number(), expand = c(0.1, 0)) + geom_label()Let us now look at the claim frequency by Brand of the car.
ggplot(Brand.summary, aes(x = reorder(Brand, Obs.Claim.Freq), y = Obs.Claim.Freq,
fill = Brand, label = percent(Obs.Claim.Freq, accuracy = 0.1))) + geom_bar(stat = "identity") +
geom_label(hjust = +1.2) + coord_flip() + guides(fill = "none") + ggtitle("Observed Claim Frequencies by Brand of the car") +
scale_x_discrete(name = "Brand") + scale_y_continuous("Observed claim Frequency",
labels = percent)The variable Gas is a categorized variable, related to the fuel of the car. We can see the different levels of a factor by using the function level in R:
levels(dataset$Gas)## [1] "Diesel" "Regular"
Gas.summary = dataset %>%
group_by(Gas) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
ggplot(Gas.summary, aes(x = Gas, y = totalExposure, fill = Gas, label = number(totalExposure))) +
geom_bar(stat = "identity") + geom_label() + guides(fill = "none") + scale_x_discrete(name = "Fuel") +
scale_y_continuous(name = "Total Exposure (in years)", labels = number)There seems to be a similar amount of Diesel and Regular gas vehicles in the portfolio. It is generally expected that Diesel have a higher claim frequency. Does this also hold on our dataset ?
ggplot(Gas.summary, aes(x = Gas, y = Obs.Claim.Freq, fill = Gas, label = percent(Obs.Claim.Freq))) +
geom_bar(stat = "identity") + geom_label() + guides(fill = "none") + scale_x_discrete(name = "Fuel") +
scale_y_continuous("Observed claim Frequency", labels = percent)The variable Region is a categorized variable, related to the region of the place of residence. We can see the different levels of a factor by using the function level in R:
levels(dataset$Region)## [1] "Aquitaine" "Basse-Normandie" "Bretagne"
## [4] "Centre" "Haute-Normandie" "Ile-de-France"
## [7] "Limousin" "Nord-Pas-de-Calais" "Pays-de-la-Loire"
## [10] "Poitou-Charentes"
What are the Exposures in each region ? What are the observed claim frequencies ?
Region.summary = dataset %>%
group_by(Region) %>%
summarize(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Region.summary| Region | totalExposure | Number.Observations | Number.Claims | Obs.Claim.Freq |
|---|---|---|---|---|
| Aquitaine | 14222.66 | 31211 | 1052 | 0.07397 |
| Basse-Normandie | 6621.74 | 10848 | 451 | 0.06811 |
| Bretagne | 27656.64 | 41986 | 1867 | 0.06751 |
| Centre | 101843.46 | 159426 | 6460 | 0.06343 |
| Haute-Normandie | 3147.22 | 8726 | 219 | 0.06959 |
| Ile-de-France | 30016.99 | 69576 | 2575 | 0.08578 |
| Limousin | 2376.00 | 4539 | 196 | 0.08249 |
| Nord-Pas-de-Calais | 11346.79 | 27111 | 939 | 0.08275 |
| Pays-de-la-Loire | 21791.75 | 38541 | 1569 | 0.07200 |
| Poitou-Charentes | 11059.29 | 18900 | 799 | 0.07225 |
We can plot a map with the observed claim frequencies and the total Exposure. We first need to obtain the shape files (which contain the borders of each administrative area.)
area <- rgdal::readOGR("shapefiles/FRA_adm1.shp", use_iconv = TRUE, encoding = "UTF-8") # From http://www.diva-gis.org/gData## OGR data source with driver: ESRI Shapefile
## Source: "C:\Users\Florian\Documents\UCLouvain\Assurance Dommage\SummerSchool\1 - Introduction & Descriptive Analysis\shapefiles\FRA_adm1.shp", layer: "FRA_adm1"
## with 22 features
## It has 9 fields
## Integer64 fields read as strings: ID_0 ID_1
# Note that the tidy function will remove the data.
area_tidy = tidy(area) # package broom
# Plot an 'empty' map
ggplot(area_tidy, aes(x = long, y = lat, group = group)) + geom_polygon(color = "black",
size = 0.1, fill = "lightgrey") + coord_equal() + theme_minimal()We are now going to include our data into the map
# First we re-include the data (that disappeared with the tidy function)
area$id <- row.names(area)
area_tidy2 <- area_tidy %>%
full_join(area@data, by = "id")
# Because of accents ...
area_tidy2$NAME_1 = stri_trans_general(str = area_tidy2$NAME_1, id = "Latin-ASCII")
# Which computed data do we want ?
data_to_add = Region.summary[, c("Region", "totalExposure", "Obs.Claim.Freq")]
# Merge it
area_tidy2 <- area_tidy2 %>%
full_join(data_to_add, by = c(NAME_1 = "Region"))
# Very important: Do not forget to sort by 'order' variable. area_tidy2 =
# area_tidy2[order(area_tidy2$order),] Easier with dplyr:
area_tidy2 = area_tidy2 %>%
arrange(order)ggplot(area_tidy2, aes(long, lat, group = group, fill = Obs.Claim.Freq)) + ggtitle("Observed Claim Frequencies") +
geom_polygon(color = "black") + scale_fill_gradient(low = "green", high = "red",
name = "Obs. Claim Freq.") + xlab("Longitude") + ylab("Latitude") + theme_void()and the exposures (on a log-scale)…
ggplot(area_tidy2, aes(long, lat, group = group, fill = log(totalExposure))) +
ggtitle("log Exposures in years") + geom_polygon(color = "black") + scale_fill_gradient(low = "green",
high = "red", name = "log Exposure") + xlab("Longitude") + ylab("Latitude") +
theme_void()The Density represents here the density of the population at the place of residence. Let us take a look at the densities in the dataset.
summary(dataset$Density)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 2 67 288 1987 1414 27000
ggplot(dataset, aes(Density)) + geom_histogram(bins = 200)Here, contrary to the age of the driver, or the age of the car, the density has lots of different values
length(unique(dataset$Density))We can compute this by using the command above, and we get 1270.
Density.summary = dataset %>%
group_by(Density) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
head(Density.summary)| Density | totalExposure | Number.Observations | Number.Claims | Obs.Claim.Freq |
|---|---|---|---|---|
| 2 | 14.07 | 18 | 1 | 0.07107 |
| 3 | 107.92 | 152 | 4 | 0.03707 |
| 4 | 39.10 | 65 | 1 | 0.02558 |
| 5 | 119.65 | 217 | 12 | 0.10029 |
| 6 | 248.92 | 399 | 11 | 0.04419 |
| 7 | 378.44 | 580 | 16 | 0.04228 |
We can plot the observed claim frequencies…
ggplot(Density.summary, aes(x = Density, y = Obs.Claim.Freq)) + geom_point()… but realize it is impossible to see a trend. One way out is to categorize the variable. We will see later (GAM) that it is possible to estimate a smooth function, which avoid the arbitrary categorization.
We can categorize the variable using the function cut.
dataset$DensityCAT = cut(dataset$Density, breaks = quantile(dataset$Density,
probs = seq(from = 0, to = 1, by = 0.1)), include.lowest = TRUE)
table(dataset$DensityCAT)##
## [2,28] (28,51] (51,91]
## 41494 41173 41330
## (91,159] (159,288] (288,562]
## 40432 41028 41408
## (562,1.16e+03] (1.16e+03,2.41e+03] (2.41e+03,4.35e+03]
## 40889 41171 42408
## (4.35e+03,2.7e+04]
## 39531
levels(dataset$DensityCAT) <- LETTERS[1:10]Then, we can apply the same strategy as above.
Density.summary = dataset %>%
group_by(DensityCAT) %>%
summarise(totalExposure = sum(Exposure), Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Density.summary| DensityCAT | totalExposure | Number.Observations | Number.Claims | Obs.Claim.Freq |
|---|---|---|---|---|
| A | 26896.70 | 41494 | 1352 | 0.05027 |
| B | 25866.44 | 41173 | 1471 | 0.05687 |
| C | 25358.47 | 41330 | 1543 | 0.06085 |
| D | 24275.29 | 40432 | 1572 | 0.06476 |
| E | 24299.85 | 41028 | 1584 | 0.06519 |
| F | 24065.32 | 41408 | 1671 | 0.06944 |
| G | 22180.06 | 40889 | 1799 | 0.08111 |
| H | 21230.08 | 41171 | 1799 | 0.08474 |
| I | 19281.92 | 42408 | 1838 | 0.09532 |
| J | 16628.45 | 39531 | 1498 | 0.09009 |
ggplot(Density.summary, aes(x = DensityCAT, y = Obs.Claim.Freq, fill = DensityCAT,
label = percent(Obs.Claim.Freq))) + geom_bar(stat = "identity") + geom_label() +
guides(fill = "none") + scale_x_discrete(name = "Density") + scale_y_continuous("Observed claim Frequency",
labels = percent)We can of course also dive into some interactions. For instance, we could analyse the effect of the car Age combined with the Fuel (Gas).
CarAge.Fuel.summary = dataset %>% group_by(CarAge, Gas) %>%
summarise(totalExposure = sum(Exposure),
Number.Observations = length(Exposure),
Number.Claims = sum(ClaimNb),
Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))## `summarise()` has grouped output by 'CarAge'. You can override using the
## `.groups` argument.
ggplot(CarAge.Fuel.summary, aes(x=CarAge,
y=Obs.Claim.Freq)) +
facet_wrap(~Gas)+
geom_bar(stat="identity") +
scale_x_continuous(name = "Age of the Car", breaks=seq(0,100,5))+
scale_y_continuous(name = "Observed Claim Frequency", labels = percent_format(accuracy = 0.01))+
theme(legend.position = 'none')We will illustrate another way to show this kind of data, by overlapping both bars.
DriverAge.Fuel.summary = dataset %>%
group_by(DriverAge, Gas) %>%
summarize(Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))## `summarise()` has grouped output by 'DriverAge'. You can override using
## the `.groups` argument.
ggplot(data = DriverAge.Fuel.summary, aes(x = DriverAge, y = Obs.Claim.Freq,
fill = Gas, color = Gas, alpha = Gas)) + geom_bar(stat = "identity", position = "identity") +
scale_x_continuous(name = "Age of the Driver", breaks = seq(0, 100, 5)) +
scale_y_continuous(name = "Observed Claim Frequency", labels = label_percent()) +
scale_colour_manual(values = c("lightblue4", "red")) + scale_fill_manual(values = c("lightblue",
"pink")) + scale_alpha_manual(values = c(0.3, 0.8)) + theme_bw()